Toward a Topological Classification of Nonadiabaticity in Chemical Reactions†

The application of topology, a branch of mathematics, to the study of electronic states in crystalline materials has had a revolutionary effect on the field of condensed matter physics. For example, the development of topological band theory has delivered new approaches and tools for characterizing the electronic structure of materials, resulting in the discovery of new phases of matter with exotic properties. In the framework of topological band theory, the crossings between energy levels of electrons are characterized by topological invariants, which predict the stability of crossings and the presence of topological boundary states. Given the frequency of energy level crossings on the potential energy surface in molecules, the applicability of these concepts to molecular systems could be of great interest for our understanding of reaction dynamics. However, challenges arise due to differing quantum mechanical descriptions of solids and molecules. Our work aims to bridge the gap between topological band theory and molecular chemistry. In particular, we propose that the Euler class, a topological invariant, can be used to categorize and analyze the distribution of nonadiabatic couplings on the potential energy surface. To exemplify this connection, we introduce a model system with two distinct regimes that are characterized by different values of the Euler class but identical potential energy surfaces. Contrary to expectations set by the Born-Oppenheimer approximation, we propose that these two regimes do not exhibit identical dynamics, due to a qualitatively distinct distribution of nonadiabatic couplings.